However there will be a large amount of data traveling between the
given servers and clients so I don't want to flood a single proxy
machine with all the traffic.
Unless you're talking about 10Gbps+ of traffic, the option that you've ruled out is the option that would be your best bet. In fact, even if you are dealing with tens of gigabits of traffic, it's likely that this is the best and safest architecture to keep sessions from being bobbled. Simply NAT your servers behind a firewall / router and be done with it. If you're worried about a single point of failure, then get an Active / Active or Active / Passive HA partner for the device.
To attempt to load balance or virtualize one IP address behind multiple NAT devices and then have said scheme manage multiple TCP/IP sessions for the servers behind the NAT array across multiple ingress and egress points will be... skull-fracturingly, eyeball-gnawingly troublesome. If you do it and manage to keep from running at full speed into a nest of bald faced hornets to distract yourself from the NAT-hurt, please write a book about it and I will order a thousand copies.